As an alternative to decomposing \(P_{uv}\)  and \(N_{uv}\)  as products of univariate non-negative functions, we tried modeling \(P_{uv}\)  and \(N_{uv}\)  as squares of linear combinations of tensor products of B-splines [1]}. That is, we let \(P_{uv}(x_u, x_v) = \left\lbrace \displaystyle \sum _{j=1}^{m} \sum _{j^{\prime }=1}^{m} s_{uj}(x_u) s_{vj^{\prime }}(x_v) \beta _{uv, jj^{\prime }}\right\rbrace ^2\)  and \(N_{uv}(x_u, x_v) = \left\lbrace \displaystyle \sum _{j=1}^{m} \sum _{j^{\prime }=1}^{m} s_{uj}(x_u) s_{vj^{\prime }}(x_v) \delta _{uv, jj^{\prime }}\right\rbrace ^2\) .
Assuming \(P_{uv}\)  and \(N_{uv}\)  to be products of univariate non-negative functions is a special case of the above model, obtained by letting \(\beta _{uv, jj^{\prime }} = \theta _{uv, 1j} \phi _{uv, 1j^{\prime }}\)  and \(\delta _{uv, jj^{\prime }} = \theta _{uv, 2j} \phi _{uv, 2j^{\prime }}.\)  This is equivalent to assuming the matrices \(\mathcal {B}_{uv} = (\beta _{uv, jj^{\prime }})_{1 \le j, j^{\prime } \le m}\)  and \(\mathcal {D}_{uv} = (\delta _{uv, jj^{\prime }})_{1 \le j, j^{\prime } \le m}\)  are rank 1. Due to the limited sample sizes and signal-to-noise ratios typical of environmental epidemiology, we found restricting the rank to 1 to be much more effective at identifying synergistic, antagonistic, or null interactions. Furthermore, the rank 1 assumption reduces the number of spline coefficients for the \(uv\) th interaction from \(2m^2\)  to \(4m\) , leading to substantial computational gains.
